If `f(x)=sin{(pi)/(2)[x]-x^(5)},1ltxlt2` and [.] denotes the greatest integer function, then `f'(5sqrt((pi)/(2)))` is equal to

Options

A. 0
B. `5(pi//2)^(4//5)`
C. `-5(pi//2)^(4//5)`
D. none of these

Answer

To solve the problem, we will follow the steps outlined in the video transcript and provide a detailed explanation for each step.

### Given:
\[ f(x) = \sin\left(\frac{\pi}{2} [x]\right) - x^5 \]
where \( 1 < x < 2 \) and \([x]\) denotes the greatest integer function.

### Step 1: Determine the value of \([x]\)
Since \( 1 < x < 2 \), we have:
\[ [x] = 1 \]
Thus, we can rewrite \( f(x) \):
\[ f(x) = \sin\left(\frac{\pi}{2} \cdot 1\right) - x^5 = \sin\left(\frac{\pi}{2}\right) - x^5 \]
Since \(\sin\left(\frac{\pi}{2}\right) = 1\), we have:
\[ f(x) = 1 - x^5 \]

### Step 2: Differentiate \( f(x) \)
Now, we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(1 - x^5) \]
Using the power rule:
\[ f'(x) = 0 - 5x^4 = -5x^4 \]

### Step 3: Evaluate \( f'(5\sqrt{\frac{\pi}{2}}) \)
Next, we need to evaluate \( f' \) at \( x = 5\sqrt{\frac{\pi}{2}} \):
\[ f'\left(5\sqrt{\frac{\pi}{2}}\right) = -5\left(5\sqrt{\frac{\pi}{2}}\right)^4 \]

### Step 4: Calculate \( \left(5\sqrt{\frac{\pi}{2}}\right)^4 \)
Calculating \( \left(5\sqrt{\frac{\pi}{2}}\right)^4 \):
\[ \left(5\sqrt{\frac{\pi}{2}}\right)^4 = 5^4 \left(\sqrt{\frac{\pi}{2}}\right)^4 = 625 \cdot \left(\frac{\pi}{2}\right)^2 = 625 \cdot \frac{\pi^2}{4} = \frac{625\pi^2}{4} \]

### Step 5: Substitute back into \( f' \)
Now substituting back into \( f' \):
\[ f'\left(5\sqrt{\frac{\pi}{2}}\right) = -5 \cdot \frac{625\pi^2}{4} = -\frac{3125\pi^2}{4} \]

### Final Answer
Thus, the value of \( f'\left(5\sqrt{\frac{\pi}{2}}\right) \) is:
\[ f'\left(5\sqrt{\frac{\pi}{2}}\right) = -\frac{3125\pi^2}{4} \]